Similarly to [1]}, we define the smoothing constraints \(c_1^k\)  and \(c_2^k\)  implicitly within the alternating updates by the simple reformulation
\(\mathbf {u}_k = \underset{\mathbf {u}}{\arg \min } \Vert  \mathbf {R}_k \mathbf {v}_k - \mathbf {u}\Vert _2^2\ s.t.\ \Vert  \mathbf {\nabla }_{\mathcal {G}} \mathbf {u}\Vert _1 \le c_1^k\iff \mathbf {u}_k = \underset{\mathbf {u}}{\arg \min } \Vert  \mathbf {\nabla }_{\mathcal {G}} \mathbf {u}\Vert _1\ s.t.\ \Vert  \mathbf {R}_k \mathbf {v}_k - \mathbf {u}\Vert _2^2 \le \hat{\sigma }^2_{\tilde{\mathbf {u}}} d \) 
